where \(v_F\)  is the Fermi velocity, \(\mathbf {p}\!\!\!=\!\!\!(p_x,p_y)\)  is the two-dimensional momentum operator, \(\mathbf {\sigma }\!=\!(\sigma _x,\sigma _y, \sigma _z)\)  are Pauli matrices acting on the atomic orbitals, \(V(r)\)  is the potential barrier, \(\Delta =166\)  meV [1]} is related to the material band gap, \(s_z = \pm 1\)  stands for the electron spin-up and spin-down, \(\tau = \pm 1\)  stands for \(k\)  and \(k^{\prime }\)  valleys, \(\lambda _{so}=75\)  meV [1]} is the splitting of the valence band due to spin-orbit coupling and \(R\)  is the dot radius. Now we'll write localized-state solutions in our system, which is characterized as a circularly symmetric quantum dot with the potential barrier \(V(r)\) , energy gap \(\Delta (r)\)  and vector potential \(\vec{A}(r)\)  associated to the magnetic flux \(\phi \)  measured in quantum unit \(\frac{h}{e}\)  given by
\(V(r) = \left\lbrace \begin{array}{cc}0, & r>R \\V, & r\le R\end{array}\right., \qquad \Delta (r) = \left\lbrace \begin{array}{cc}0, & r>R \\\Delta , & r\le R\end{array}\right., \qquad \vec{A}(r)=\left\lbrace \begin{array}{ll}\frac{h}{e}\frac{\phi }{2\pi r}\vec{e}_{\theta }, & r< R \\0, & r>R \\\end{array}\right.\) 
